Question: (continuation) The value of θ is called the Feature Completeness Index (FCI). The larger the FCI, the better the product. We want to estimate the FCI. Suppose that n raters will provide scores for the product. We construct a model for the n scores by proposing a random variable \[{{X}_{i}}\] for the score of each rater. We assume the Random Sample model, with the above probability density function (from problem 1) for each \[{{X}_{i}}\].

a) Suppose the model proposes that each realization \[{{x}_{i}}\] “should be” equal to \[\frac{\theta }{\theta +1}\]. Given realizations \[{{x}_{1}},{{x}_{2}},...,{{x}_{n}}\], what is the least-squares estimate of the FCI?

b) Given realizations \[{{x}_{1}},{{x}_{2}},...,{{x}_{n}}\], what is the maximum likelihood estimate of the FCI?

c) What is \[p\lim (\bar{X})\] ?

d) Provide values for a and b so that \[a\bar{X}-b\] has an asymptotic standard normal distribution.